Abstract

We investigate the modified Mellin transform on certain function space of generalized functions. We first obtain the convolution theorem for the classical and distributional modified Mellin transform. Then we describe the domain
and range spaces where the extended modified transform is well defined. Consistency, convolution, analyticity, continuity, and sufficient theorems for the proposed transform have been established. An inversion formula is also obtained and many properties are given.

1. Introduction

The Mellin transform of a suitably restricted function over was defined on some strip in the complex plane [1], where many of the transform properties are obtained by applying change of variables to various properties of the Laplace transformation. The Mellin transform is extended to distributions in [1] and to Boehmians in [2].

By combining Fourier and Mellin transforms, the obtained transform is called Fourier-Mellin transform which has many applications in digital signals, image processing, and ship target recognition by sonar system and radar signals as well.

The modified Mellin transform of a suitably restricted function over was introduced by [3]
as a scale-invariant transform. Then, as earlier, combining the modified Mellin transform with the Fourier transform gives the Fourier-modified Mellin transform [3, Equation 16].

The Mellin-type convolution product of two functions and is given by
From [1] it has been noted that
Utilizing the Mellin-type convolution product the following theorem is essential for our next investigations.

Theorem 1 (convolution theorem). Let be the Lebesgue space of integrable functions and ; then
where and are the Mellin-type and Mellin transforms of and , respectively.

Proof. By the definition of the Mellin and modified Mellin transforms we have
Employing Fubinis theorem, then the substitution together with simple computations establishes that
Hence the theorem is proved.

2. Modified Mellin Transform of Distribution

Let be the space of smooth functions over such that [1]
is finite, , where
and .

Then is linear space under addition and multiplication by complex numbers. can also be generated by the multinorms which turns to be a countably multinormed space.

Denote by the complete strong dual space of ; then it is assigned the weak topology. Let be the space of smooth functions over ; then for any , is dense in and the topology assigned to is stronger than that induced on by and is identified with a subspace of .

The straightforward conclusion is that the kernel function of is a member of for .

This usually leads to the following definition: let ; then the distributional transform of is defined as
where and .

Theorem 2 (analyticity theorem). Let ; then is analytic and
where is nonnegative integer and .

Therefore, denote by the space of test functions of bounded support over ; then the convolution product of and can be given as
where .

3. Boehmians

Let a group and a subgroup of . We assume that to each pair of elements and is assigned the product such that(1)if , then and ;(2)if and , then ;(3)if , and , then
Let be a family of sequences from such that(1)if , and , then , for all ;(2)if , then . Elements of will be called delta sequences.

Consider the class of pair of sequences defined by
for each .

An element is called a quotient of sequences, denoted by , if , for all .

Two quotients of sequences and are said to be equivalent, , if , for all .

The relation is an equivalent relation on . The equivalence class containing is denoted by . These equivalence classes are called Boehmians. The space of all Boehmians is denoted by .

The sum of two Boehmians and multiplication by a scalar can be defined in a natural way
, space of complex numbers.

The operation and the differentiation are defined by and .

The operation can be extended to as follows. If and , then

δ-Convergence. A sequence of Boehmians in is said to be convergent to a Boehmian in , denoted by , if there exists a delta sequence such that

The following is equivalent for the statement of -convergence: in if and only if there is and such that , and for each , as in .

A sequence of Boehmians in is said to be convergent to a Boehmian in , denoted by , if there exists a such that , for all , and as in . See [2, 5–15].

4. Modified Mellin Transform of Boehmian

In this section we discuss the modified Mellin transform on spaces of Boehmians. Consider the group and as a subgroup of . Let be the operation between and and the set of delta sequences given by [2](1), for all ;(2), for all , for some ;(3)supp , for all for some with , as . Let be the Boehmian space obtained from and ; then will serve as the domain space of .

Our next objective is to construct a range space, say , for

Let
For , define
We have the following theorem.

Theorem 3. Let and , then .

Proof. Let and belong to and ; respectively. Then there are and such that and , respectively. Therefore, by the convolution theorem and (19) we get that
Since , it follows .Hence we have proved the theorem.

Theorem 4. Let ; then .

Proof. By definition of we can find such that and .Therefore, by [1],
But since , we get . Thus we have the theorem.

Theorem 5. Let and ; then and .

Proof.
Is straightforward.

Theorem 6. Let in ; then as in .

Proof.
Can easily be checked.

Theorem 7. Let and ; then as .

Proof. By (19) we have
where and are such that and , for all .Since as on compact subsets of , (22) implies that , for all , as . Hence we obtain the theorem.

Theorem 8. Let ; then .

Let ; then taking into account the fact that , since , this theorem follows.

The Boehmian space is therefore constructed.

In addition, scalar multiplication, differentiation, and the operation in are defined similar to that of usual Boehmian spaces.

Proof. For every , let be its representative in ; then , where , for all . Then it is clear that is independent of the representative, for all .Therefore
which is the representative of in .Hence we have the proof.

Theorem 12 (necessity theorem). Let ; then the necessary and sufficient condition that is to be in the range of is that belongs to range of for every .

Proof. Let be in the range of then of course belongs to the range of , for all .To establish the converse, let be in the range of , for all . Then there is such that .Since ,
for all . Therefore,
where and .The fact that is injective, , implies that .Thus is quotient of sequences in . Hence, and
Hence the theorem is proved.

Theorem 13 (generalized convolution theorem). Let and ; then

Proof. Assume that the requirements of the theorem satisfy for some and ; then using Definition 9 and the operation yields
Therefore
This completes the proof.

Theorem 14. The extended modified Mellin transform is bijective.

Proof. Assume ; then it follows from the concept of quotients of sequences that . Therefore, . The property that is one to one implies . Therefore,
Next to establish that is onto, let be arbitrary; then for every . Hence are such that , for all .Hence, the Boehmian belongs to and satisfies
This completes the proof of the theorem.

Now we introduce as the inverse transform of , where
for every .

Theorem 15. Let and , then

Proof. We prove (35) and omit the proof of (36) due to its similarity. Given and such that then employing (34) yields
Using (19) gives
Hence the convolution theorem gives
Thus
Proof of the second part is similar.This completes the proof of the theorem.

Theorem 16. and are continuous with respect to -convergence.

Proof. Let in as ; then we establish that as . Let and be in such that
and as for every .The continuity of implies as . Thus,
as in . This proves continuity of .Next, let as ; then we have and for some , where as . Hence
as in . That is,
as .Hence
as .That is,
as . This completes the proof.

Theorem 17. and are continuous with respect to -convergence.

Proof. Let in as . Then, we find and such that and as . Therefore
Hence, as in .Therefore
Hence, as .Proof of the second part is analogous. Detailed proof is as follows.Finally, let in as ; then we can find such that and as for some .Next, we have
Thus, by (34) we get
as in .Therefore
as .Thus, we have
as in .This completes the proof of the theorem.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors acknowledge that this research was partially supported by Universiti Putra Malaysia under the ERGS 1-2013/5527179.